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Given that $ \displaystyle \lim_{x \to 2}(5x - 7) = 3 $, illustrate Definition 2 by finding values of $ \delta $ that correspond to $ \varepsilon = 0.1 $, $ \varepsilon = 0.05 $, and $ \varepsilon = 0.01 $.

$\delta=0.02$ and $\delta=0.01$ and $\delta=0.002$

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This is problem number fourteen of the Stuart Calculus eighth edition, section two point four. Given that the limit is ex A purchase to whom? Of the quantity Primex Money seven is equal to dream. Illustrate definition to buy pining values of Delta that corresponded to absolute is equal to zero point one absolutely is equal to zero point five and a plan is equal to zero point zero one. So we were called the definition definition, too. Foreign limits, his expression, say of the function equals to hell that this limit equal zone if, for every absolutely greater than zero we can find okay, tell tell greater than zero that if the difference between ex inane it's Listen, doctor, then the difference between the function is Listen, Absalon. All right, so we're going to work with this definition. In fact, we're going to begin Ah, with this second inequality. So they absolutely you of the function minus how less than absolute. We're gonna plug in the function X minus seven minus L, which is three in this case. Remain current here. There should be absolutely, and our Alice three the absolutely the quantity of the function minus a limit is. Listen, Absalon, here we simplify five X minus seven minus three. It's the same. It's fun. Experience. Ten. This quantity in the absolutely sciences. Liston Absalon We could do some simplifying here. Fact already. Fine. Listen up Salon And then we're going to do is wear enough divided by five and pause it here. Okay, The reason we pause here is because this is similar to the condition for Delta because an hour limit exit purchased two is equal to. So our delta is found through this inequality the absolutely of thie quantity X minus two last on Delta Notice that working with the second inequality, we were able to determine this very similar inequality. The absolutely of the quantity explains to less than absolutely fine. Therefore, we stayed. Delta is equal to in this valley. These two sides have to be equal, or at least is the case for us. Since we're looking for a limiting really for Delta and we're going to use each value of Absalon to determine three different doctors. The first Delta we'll be the first value of Absalom. Is there a white one divided by five. And for us, that is equal to zero point zero two. The second Delta is going to be equal to the second. Absalon is there when there are five, the right of it. Fine. This is equal to zero point zero one in the front of the whole town is equal to the turn. Absalon zero point zero one to find my find, which is equal to zero point zero zero two. And when we did as we and provided a argument for this limit through definition to finding three values. Delta's for three different salons, again following the instructions of Tel toe our definition, too.